Let's break down the problem step-by-step. ## **Problem Restatement** A force **P** acts at point **E** with a direction. You are asked to determine the **perpendicular distance** between the line of action of force **P** and the axis of pipe **OC** (which is along the x-axis). ## **Step 1: Find the Coordinates of Points** Let’s assign coordinates to the points based on the given dimensions: - **O**: (, , ) - **C**: (10, 7.5, ) - **E**: (10, 7.5, 7.5) - **P** acts through **E** in the direction from **E** to **F**. Let’s find coordinates for **F**: - **F**: (, 7.5, 7.5) ## **Step 2: Find Direction Vector of Pipe OC** \[ \overrightarrow{OC} = \langle 10-, 7.5-, - \rangle = \langle 10, 7.5, \rangle \] ## **Step 3: Find Direction Vector of Line of Action of P (EF)** \[ \overrightarrow{EF} = \langle -10, 7.5-7.5, 7.5-7.5 \rangle = \langle -10, , \rangle \] But the force acts from **E** towards **F**, so the direction vector is: \[ \mathbf{d}_P = \langle -10, , \rangle \] ## **Step 4: Find Perpendicular Distance Between Skew Lines (Formula)** The perpendicular distance \(d\) between two skew lines (not parallel, not intersecting) can be found by: \[ d = \frac{|\left( \mathbf{r}_2 - \mathbf{r}_1 \right) \cdot (\mathbf{u}_1 \times \mathbf{u}_2)|}{|\mathbf{u}_1 \times \mathbf{u}_2|} \] Where: - \(\mathbf{r}_1\) is a point on line 1 (**O**), - \(\mathbf{u}_1\) is the direction vector of line 1 (**OC**), - \(\mathbf{r}_2\) is a point on line 2 (**E**), - \(\mathbf{u}_2\) is the direction vector of line 2 (**EF**). Let’s assign: - \(\mathbf{r}_1 = (, , )\) - \(\mathbf{u}_1 = (10, 7.5, )\) - \(\mathbf{r}_2 = (10, 7.5, 7.5)\) - \(\mathbf{u}_2 = (-10, , )\) ## **Step 5: Find \(\mathbf{u}_1 \times \mathbf{u}_2\)** \[ \mathbf{u}_1 \times \mathbf{u}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 10 & 7.5 & \\ -10 & & \\ \end{vmatrix} \] \[ = \mathbf{i}(7.5 \cdot - \cdot ) - \mathbf{j}(10 \cdot - \cdot -10) + \mathbf{k}(10 \cdot - 7.5 \cdot -10) \] \[ = \mathbf{i}() - \mathbf{j}() + \mathbf{k}( + 75) \] \[ = (, , 75) \] ## **Step 6: Find \(\mathbf{r}_2 - \mathbf{r}_1\)** \[ \mathbf{r}_2 - \mathbf{r}_1 = (10, 7.5, 7.5) - (, , ) = (10, 7.5, 7.5) \] ## **Step 7: Dot Product \((\mathbf{r}_2 - \mathbf{r}_1) \cdot (\mathbf{u}_1 \times \mathbf{u}_2)\)** \[ (10, 7.5, 7.5) \cdot (, , 75) = 10 \cdot + 7.5 \cdot + 7.5 \cdot 75 = 562.5 \] ## **Step 8: Magnitude of Cross Product** \[ |\mathbf{u}_1 \times \mathbf{u}_2| = |(, , 75)| = 75 \] ## **Step 9: Calculate Perpendicular Distance** \[ d = \frac{|562.5|}{75} = 7.5 \] ## **Step 10: Final Answer** \[ \boxed{7.5 \text{ in.}} \] --- ### **Summary Table** | Step | Calculation | Value | |-------------|------------------------------|---------------| | Direction OC| (10, 7.5, ) | | | Direction EF| (-10, , ) | | | Cross Prod | (, , 75) | 75 | | r2 - r1 | (10, 7.5, 7.5) | | | Dot Product | 562.5 | | | Distance | \( \frac{562.5}{75} = 7.5 \) | 7.5 in | --- ## **Final Answer** \[ \boxed{7.5 \text{ in.}} \] This is the perpendicular distance between the line passing through OC and the line of action of P at E.